Amplitude dynamics of the charge density wave in ${\mathrm{LaTe}}_3$: Theoretical description of pump-probe experiments

We formulate a dynamical model to describe a photoinduced charge density wave (CDW) quench transition and apply it to recent multiprobe experiments on ${\mathrm{LaTe}}_3$ [A. Zong et al., Nat. Phys. 15, 27 (2019)]. Our approach relies on coupled time-dependent Ginzburg-Landau equations tracking two order parameters that represent the modulations of the electronic density and the ionic positions. We aim at describing the amplitude of the order parameters under the assumption that they are homogeneous in space. This description is supplemented by a three-temperature model, which treats separately the electronic temperature, temperature of the lattice phonons with stronger couplings to the electronic subsystem, and temperature of all other phonons. The broad scope of available data for ${\mathrm{LaTe}}_3$ and similar materials as well as the synergy between different time-resolved spectroscopies allow us to extract model parameters. The resulting calculations are in good agreement with ultrafast electron diffraction experiments, reproducing qualitative and quantitative features of the CDW amplitude evolution during the initial few picoseconds after photoexcitation.

Figure 1

Constraints on the allowed values of the TDGL parameters ${\tau }_0$ and ${\omega }_0$ formulated in Sec. 3a. The area above the solid red curve is physically inaccessible because it corresponds to ${\gamma }_y({\omega }_0,{\tau }_0)<0$. The available parameter space to the left and below the solid curve hosts two regimes of small oscillations, adiabatic (above the dashed line) and nonadiabatic (below the dashed line), see Sec. 3a and Appendix pp2. These two regimes are exemplified by two points, (a) and (b), for which the mode softening is illustrated in Fig. 11. The dashed line is determined by equation $\mathcal{D}=0$, where $\mathcal{D}$ is defined by Eq. (B9). The parameters ${\omega }_0$ and ${\tau }_0$ for ${\mathrm{LaTe}}_3$ are assumed to represent the adiabatic regime.

Figure 2

Typical example of the time evolution of electronic temperature $T_{\mathrm{e}}$ (solid red line), temperature of hot phonons $T_{\mathrm{L}2}$ (dashed blue line), and temperature of cold phonons $T_{\mathrm{L}1}$ (dash-dotted black line). The curves are computed numerically on the basis of Eqs. (16, 17, 18) with the initial value of $T_{\mathrm{e}}$ corresponding to the photoexcitation density $F=2\times {10}^{20}{\mathrm{cm}}^{-3}$ and with parameters given in Table 1.

Figure 3

Electronic temperature dynamics for different photoexcitation densities $F$, as described by the three-temperature model. The in-plot legend explains the correspondence between $F$ and the curves. Horizontal dash-dotted line marks the CDW transition temperature $T_{\mathrm{c}}=670$ K. The crossing of this line with an individual temperature evolution curve $T_{\mathrm{e}}\left(t\right)$ defines time ${\tau }_{\mathrm{c}}\left(F\right)$ introduced by Eq. (29).

Figure 4

Time scales ${\tau }_{\mathrm{e}}$, Eq. (28), and ${\tau }_{\mathrm{c}}$, Eq. (29), as functions of photoexcitation density $F$. Crossed points correspond to the quasiparticle time ${\tau }_{\mathrm{R}}$, extracted from the transient reflectivity measurements [8].

Figure 5

Time evolution of the CDW order parameters. Four panels correspond to different photoexcitation densities $F$ (shown in each panel in units of ${10}^{20}{\mathrm{cm}}^{-3})$. Dashed blue lines show the time dependence of the electronic CDW order $x^2\left(t\right)$ normalized to its prepulse value $x^2\left(0^{{}^{-}}\right)$. Similarly, solid red lines show the time evolution of the lattice CDW order represented as $y^2\left(t\right)/y^2\left(0^{{}^{-}}\right)$. Vertical dashed lines mark $t={\tau }_{\mathrm{c}}\left(F\right)$ defined by Eq. (29).

Figure 6

Time evolution of the lattice order parameter $y^2\left(t\right)/y^2(t<0)$—see Fig. 5—filtered with Gaussian function in Eq. (31) in order to mimic finite resolution in the UED experiment, for several values of $F$.

Figure 7

Static electron diffraction pattern along ($30L)$. The line cut is obtained by integrating the colored strip along the $H$ direction. The measured diffraction is a two-dimensional slice in the three-dimensional reciprocal space. The bright yellow spots are Bragg peaks while arrows mark the CDW superlattice peaks. Figure reproduced from Ref. [8].

Figure 8

Time evolution of the Bragg peaks intensity $I\left(t\right)$ for different excitation densities. Solid lines are obtained from Eq. (34); dots represent the experimental data [8]. The lines and the data points for different excitation densities are vertically displaced for clarity.

Figure 9

Comparison of the simulated dynamics with the UED data for the CDW peak. Two panels correspond to two different photoexcitation densities: $F=9.4\times {10}^{19}$ (top) and $2.8\times {10}^{20}{\mathrm{cm}}^{-3}$ (bottom). Quantity $(y^2*g)\left(t\right)$, representing simulated filtered dynamics of the order parameter (see Fig. 6), is shown by dashed (red) curve. Experimentally obtained [8] partially integrated UED intensity $I_{2\mathrm{D}}^{\mathrm{CDW}}$, Eq. (37), is shown by solid (cyan) curves. The data points [8] for the CDW correlation length $\xi$ are shown as (blue) dots. For larger $F$ (bottom), the CDW peak disappears for 1 ps $\lesssim t\lesssim 3.5$ ps, consequently, the data points for $\xi$ are absent in this interval. To account for the theoretically unknown phase dynamics, we multiply $(y^2*g)$ by $\xi$ [as in Eq. (40)]. The resulting dependence is shown by solid (green) line. For both $F$'s, the agreement between $I_{2\mathrm{D}}^{\mathrm{CDW}}$ and $(y^2*g)\xi$ is quite notable.

Figure 10

Sketch of the stages of nonequilibrium time evolution in the course of a photoexcited phase transition in ${\mathrm{LaTe}}_3$. The characteristic times indicated in the figure are the estimates made in Sec. 5 for $F=3\times {10}^{20}{\mathrm{cm}}^{-3}$.

Figure 11

Temperature dependence of the amplitude mode frequency ${\omega }_{\mathrm{AM}}$ and damping parameter ${\gamma }_{\mathrm{AM}}$ obtained by solving Eqs. (B6) and (B7). (a) Adiabatic regime: the parameters in Eq. (B7) are ${\omega }_0/\left(2\pi \right)=3.1$ THz, ${\tau }_0=20$ fs, $\zeta \approx 1.1$, and ${\gamma }_y\approx 0.04$. These parameter values were used in our simulations, see Table 1. They correspond to point (a) marked in Fig. 1. In a small temperature range around $T_{\mathrm{c}}$, the amplitude oscillation mode turns into two overdamped modes with ${\omega }_{\mathrm{AM}}=0$ and unequal values of ${\gamma }_{\mathrm{AM}}$ represented by two split dashed lines. (b) Nonadiabatic regime: the parameters in Eq. (B7) are ${\omega }_0/\left(2\pi \right)=2.6$ THz, ${\tau }_0=40$ fs, $\zeta \approx 0.5$, and ${\gamma }_y\approx 0.033$. This choice of parameters corresponds to point (b) marked in Fig. 1.

Figure 12

Sketch of DOS in the CDW state. It represents the vicinity of a Fermi surface that has both metallic and gaped regions. The gapped regions are responsible for the suppression of the DOS at ${\epsilon }_F=0$ and the appearance of the maxima around $\left|\varepsilon \right|\sim \mathrm{\Delta }\sim 350$ meV.

Figure 13

Dashed (red) line represents temperature dependence of the electronic heat capacity for the DOS shown in Fig. 12. Solid (blue) line corresponds to our approximation Eq. (23).

Figure 14

Estimating electronic temperature $T_{\mathrm{e}}$ after a photoexcitation. (a) Fermi surface map before arrival of a laser pulse $\left(t<t_0\right)$. Intensities are integrated over $\pm 10$ meV around the Fermi energy ${\epsilon }_F$. Dashed curves represent calculated Fermi surface based on a tight-binding model [91], in the absence of the CDW order. The arrow marks the energy-momentum cut through the ungapped part of the Fermi surface displayed in (b). In (b), the band dispersions are shown at two representative pump-probe time delays: before the laser pulse (left) and 250 fs after the pulse arrival (right). The energy-distribution curves (EDCs) are obtained by integrating over a window $\mathrm{\Delta }k=0.05{\AA }^{-1}$ at the momentum indicated by the arrow. Blue curve is a fit by Eq. (C2) to a part of the EDC. The Fermi energy is indicated by the dashed line. Data in (a) and (b) were obtained at a photoexcitation density of $3.31\times {10}^{20}{\mathrm{cm}}^{-3}$. (c) Electronic temperature $T_{\mathrm{e}}$ plotted as a function of the pump-probe delay for the three photoexcitation densities indicated in the plot legend. Curves are the fits to a single-exponential decay model [8]. Error bars represent one standard deviation of the fits.